A particle with velocity $v(t)=e^t$, where $t$ is time in seconds, moves in a straight line. How far does the particle move from $t=0$ to $t=2$ seconds? Choose 1 answer: Choose 1 answer: (Choice A) A $1$ unit (Choice B) B $e^2$ units (Choice C) C $e^2-1$ units (Choice D) D $e^2-e$ units
Answer: The definite integral $ \int_a^b |v(t)| \,dt$ gives the total distance traveled by a particle over the interval $[a,b]$. In this case, $ \int_0^2 |e^t| \,dt$ represents the total distance traveled by the particle from $t=0$ to $t=2$ seconds. Since $e^t$ is always positive, we can rewrite the integral: $ \int_0^2 e^t \,dt$ We can now evaluate the integral: $\begin{aligned} \int_0^2 e^t \,dt&=\Big[e^t\Big]_0^2\\ \\ \\ &=e^2-e^0\\ \\ &=e^2-1\\ \end{aligned}$ The answer: $e^2-1$ units